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If we add this relation to equation (9.14) restricted to Vh , we see that the
weak solution u ∈ V © H k+1 („¦) satis¬es the following variational equation:
for all vh ∈ Vh ,
ah (u, vh ) = f, vh h

where
δK ’µ∆u + c · ∇u + ru, „ (vh )
ah (u, vh ) := a(u, vh ) + ,
0,K
K∈Th

f, vh := f, v + δK f, „ (vh ) .
h 0 0,K
K∈Th

Then the corresponding discretization reads as follows:
Find uh ∈ Vh such that
for all vh ∈ Vh .
ah (uh , vh ) = f, vh (9.18)
h

Corollary 9.2 Suppose the problems (9.14) and (9.18) have a solution u ∈
V © H k+1 („¦) and uh ∈ Vh , respectively. Then the following error equation
is valid:
ah (u ’ uh , vh ) = 0 for all vh ∈ Vh . (9.19)
9.2. Streamline-Di¬usion Method 377

In the streamline-di¬usion method (sdFEM), the mapping „ used in (9.18)
is chosen as „ (vh ) := c · ∇vh .
Without going into details, we mention that a further option is to set
„ (vh ) := ’µ∆vh + c · ∇vh + rvh . This results in the so-called Galerkin/least
squares“FEM (GLSFEM) [54].
Especially with regard to the extension of the method to other ¬nite ele-
ment spaces, the discussion of how to choose „ and δK is not yet complete.

Interpretation of the Additional Term in the Case of Linear Ele-
ments
If the ¬nite element spaces Vh are formed by piecewise linear functions (i.e.,
in the above de¬nition (9.15) of Vh we have k = 1), we get ∆vh |K = 0 for
all K ∈ Th . If in addition there is no reactive term (i.e., r = 0), the discrete
bilinear form is


µ∇uh ·∇vh dx+ c · ∇uh , vh 0 + δK c · ∇uh , c · ∇vh
ah (uh , vh ) = .
0,K
„¦ K∈Th


Since the scalar product appearing in the sum can be rewritten as
c · ∇uh , c · ∇vh 0,K = K (ccT ∇uh ) · ∇vh dx , we obtain the following
equivalent representation:


(µI + δK ccT )∇uh · ∇vh dx + c · ∇uh , vh
ah (uh , vh ) = .
0
K
K∈Th


This shows that the additional term introduces an element-dependent extra
di¬usion in the direction of the convective ¬eld c (cf. also Exercise 0.3),
which motivates the name of the method. In this respect, the streamline-
di¬usion method can be understood as an improved version of the full
upwind method, as seen, for example, in (9.6).

Analysis of the Streamline-Di¬usion Method
To start the analysis of stability and convergence properties of the
streamline-di¬usion method, we consider the term ah (vh , vh ) for arbitrary
vh ∈ Vh .
As in Section 3.2.1, the structure of the discrete bilinear form ah allows
us to derive the estimate

ah (vh , vh ) ≥ µ|vh |2 +r0 vh ’µ∆vh + c · ∇vh + rvh , c · ∇vh
2
0+ δK .
1 0,K
K∈Th


Furthermore, neglecting for a moment the second term in the sum and
using the elementary inequality ab ¤ a2 + b2 /4 for arbitrary a, b ∈ R, we
378 9. Discretization of Convection-Dominated Problems

get

δK ’µ∆vh + rvh , c · ∇vh 0,K
K∈Th

¤ ’µ |δK | ∆vh , |δK | c · ∇vh
0,K
K∈Th

|δK | rvh , |δK | c · ∇vh
+
0,K

¤ µ2 |δK | ∆vh + |δK | r
2 2 2
vh
0,K 0,∞,K 0,K
K∈Th
|δK |
c · ∇vh 2
+ .
0,K
2
By means of the inverse inequality (9.17) it follows that

c2
δK ’µ∆vh + rvh , c · ∇vh ¤ µ2 |δK | 2 |vh |1,K
inv 2
0,K
hK
K∈Th K∈Th
|δK |
+|δK | r c · ∇vh 2
2 2
vh + .
0,K
0,∞,K 0,K
2
Putting things together, we obtain
c2
ah (vh , vh ) ≥ µ ’ µ |δK | inv |vh |2
2 2
1,K vh 0,K
h2
K
K∈Th
|δK |
+ r0 ’ |δK | r + δK ’ c · ∇vh
2 2
.
0,∞,K 0,K
2
The choice
h2
1 r0
¤ min K
0 < δK 2 , r2 (9.20)
2 µcinv 0,∞,K

leads to
µ r0 1
ah (vh , vh ) ≥ |vh |2 + δK c · ∇vh
2 2
vh + .
1 0 0,K
2 2 2
K∈Th

Therefore, if the so-called streamline-di¬usion norm is de¬ned by
1/2

δK c · ∇v v∈V ,
µ|v|2 + r0 v 2 2
v := + ,
sd 1 0 0,K
K∈Th

then the choice (9.20) implies the estimate
1
¤ ah (vh , vh ) for all vh ∈ Vh .
2
vh (9.21)
sd
2
9.2. Streamline-Di¬usion Method 379

Obviously, the streamline-di¬usion norm · sd is stronger than the
µ-weighted H 1 -norm (9.10); i.e.,

min{1, r0 } v µ ¤ v sd for all v ∈ V .

Now an error estimate can be proven. Since estimate (9.21) holds only on
the ¬nite element spaces Vh , we consider ¬rst the norm of Ih (u) ’ uh ∈ Vh
and make use of the error equation (9.19):
1
Ih (u) ’ uh ¤ ah (Ih (u) ’ uh , Ih (u) ’ uh ) = ah (Ih (u) ’ u, Ih (u) ’ uh ) .
2
sd
2
In particular, under the assumption u ∈ V © H k+1 („¦) the following three
estimates are valid:

∇(Ih (u) ’ u) · ∇(Ih (u) ’ uh ) dx ¤ µ |Ih (u) ’ u|1 Ih (u) ’ uh
µ sd
„¦

¤ cint µ hk |u|k+1 Ih (u) ’ uh ,
sd



[c · ∇(Ih (u) ’ u) + r(Ih (u) ’ u)](Ih (u) ’ uh ) dx
„¦

(r ’ ∇ · c)(Ih (u) ’ u)(Ih (u) ’ uh ) dx
=
„¦

’ (Ih (u) ’ u) c · ∇(Ih (u) ’ uh ) dx
„¦

¤ r’∇·c Ih (u) ’ u Ih (u) ’ uh
0,∞ 0 0

+ Ih (u) ’ u c · ∇(Ih (u) ’ uh )
0 0
±
 1/2

¤ Ih (u) ’ u 2
C
 0,K
K∈Th

1/2 
’1
δK Ih (u) ’ u Ih (u) ’ uh
2
+ sd

0,K
K∈Th
1/2
’1
¤ 1 + δK h2 |u|2 Ih (u) ’ uh
Chk ,
sd
K k+1,K
K∈Th

and

δK ’µ∆(Ih (u) ’ u) + c · ∇(Ih (u) ’ u)
K∈Th

+ r(Ih (u) ’ u), c · ∇(Ih (u) ’ uh ) (9.22)
0,K
380 9. Discretization of Convection-Dominated Problems

¤ δK µhk’1 + c k+1
k
cint 0,∞,K hK +r 0,∞,K hK
K
K∈Th
— |u|k+1,K δK c · ∇(Ih (u) ’ uh ) 0,K


1/2
2
¤C |u|2 Ih (u) ’ uh
δK µhk’1 + hk + hk+1 sd .
K k+1,K
K K
K∈Th

Condition (9.20), which was already required for estimate (9.21), implies
that
h2
¤ 2K ,
µδK
cinv
and so the application to the ¬rst term of the last bound leads to

δK ’ µ∆(Ih (u) ’ u) + c · ∇(Ih (u) ’ u)
K∈Th
+ r(Ih (u) ’ u), c · ∇(Ih (u) ’ uh ) 0,K

1/2

¤ Ch δK ] |u|2 Ih (u) ’ uh
k
[µ + .
sd
k+1,K
K∈Th

Collecting the estimates and dividing by Ih (u) ’ uh sd , we obtain the
relation
1/2
h2
Ih (u) ’ uh ¤ Chk µ + K + h2 + δK |u|2 .
sd K k+1,K
δK
K∈Th

Finally, the terms in the square brackets will be equilibrated with the help
of condition (9.20). We rewrite the µ-dependent term in this condition as
h2 2
K
=2 PeK hK
2
µcinv cinv c ∞,K

with
c ∞,K hK
PeK := . (9.23)

This local P´clet number is a re¬nement of the de¬nition (9.4).
e
The following distinctions concerning PeK are convenient:
PeK ¤ 1 and PeK > 1 .
In the ¬rst case, we choose
h2 2
= δ1 K ,
δK = δ0 PeK hK δ0 = δ1 ,
µ c ∞,K
9.2. Streamline-Di¬usion Method 381

with appropriate constants δ0 > 0 and δ1 > 0, respectively, which are
independent of K and µ. Then we have
h2 1 2PeK
hK ¤ C(µ + hK ) ,
+ h2 + δK = µ + h2 + δ1
K
µ+ 1+
K K
δK δ1 c 0,∞,K
where C > 0 is independent of K and µ. In the second case, it is su¬cient to
choose δK = δ2 hK with an appropriate constant δ2 > 0 that is independent
of K and µ. Then
δ2 c 0,∞,K h2
δ2 K
δK = PeK hK =
PeK 2PeK µ
and
h2 1
+ δ2 hK + h2 ¤ C(µ + hK ) ,
µ + K + h2 + δK = µ +
K K
δK δ2
with C > 0 independent of K and µ. Note that in both cases the constants
can be chosen su¬ciently small, independent of PeK , that the condi-
tion (9.20) is satis¬ed. Now we are prepared to prove the following error
estimate.
Theorem 9.3 Let the parameters δK be given by
±
 h2
δ1 K , PeK ¤ 1 ,
δK =
 δ hµ , Pe > 1 ,
2K K

where δ1 , δ2 > 0 do not depend on K and µ and are chosen such that
condition (9.20) is satis¬ed. If the weak solution u of (9.14) belongs to
H k+1 („¦), then


u ’ uh sd ¤ C µ + h hk |u|k+1 ,

where the constant C > 0 is independent of µ, h, and u.

Proof: By the triangle inequality, we get
u ’ uh ¤ u ’ Ih (u) + Ih (u) ’ uh sd .
sd sd

An estimate of the second addend is already known. To deal with the ¬rst
term, the estimates of the interpolation error (9.16) are used directly:


u ’ Ih (u) 2
sd

µ|u ’ Ih (u)|2 + r0 u ’ Ih (u) δK c · ∇(u ’ Ih (u))
2 2
= +
1 0 0,K
K∈Th
2(k+1)
¤ |u|2
c2 µh2k + r0 hK 2 2k
+ δK c 0,∞,K hK
int K k+1,K
K∈Th
382 9. Discretization of Convection-Dominated Problems

¤ µ + h2 + δK |u|2
k+1,K ¤ C(µ + h)hK |u|k+1 .
Ch2k 2k 2
K K
K∈Th

2

Remark 9.4 (i) In the case of large local P´clet numbers, we have
e
µ¤ 1 c ∞,K hK and thus
2

1/2

u ’ uh hK c · ∇(u ’ uh ) ¤ Chk+1/2 |u|k+1 .
2
+ δ2
0 0,K
K∈Th


So the L2 -error of the solution is not optimal in comparison with the
estimate of the interpolation error

u ’ Ih (u) ¤ Chk+1 |u|k+1 ,
0


whereas the L2 -error of the directional derivative of u in the direction

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